### The period of 1/k

The meaning of the term ‘period’ is as follows. Take the case of 1/11 = 0.090909. Here the two digits 0 and 9 are repeated forever. So the period of 1/11 is 2. Another example is 1/7 = 0.142857142857… Here the 6 digits 142857 are repeated forever. So the period of 1/7 is 6. (There are lots of interesting properties for the number 1/7. For more details see this link)

There is a Theorem about the period: The period of 1/k for integer k is always < k. I have a proof for this below.